If x=5+35−3 and y=5−35+3 , then the value of x2+y2+xy is

Given x=5+35−3,y=5−35+3x+y=(5+3)2+(5−3)2(5)2−(3)2=2(5+3)2=8xy=5+35−35−35+3=1∴x2+y2+xy=(x+y)2−xy=82−1=63

If x=5+35−3 and y=5−35+3 , then the value of x2+y2+xy is

Given x=5+35−3,y=5−35+3x+y=(5+3)2+(5−3)2(5)2−(3)2=2(5+3)2=8xy=5+35−35−35+3=1∴x2+y2+xy=(x+y)2−xy=82−1=63

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If $x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}$ and $y = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$ , then the value of $x^{2} + y^{2} + xy$ is

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Detailed Solution

Given $x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}, y = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$
$\begin{array}{l} x + y = \frac{(\sqrt{5} + \sqrt{3})^{2} + (\sqrt{5} - \sqrt{3})^{2}}{(\sqrt{5})^{2} - (\sqrt{3})^{2}} = \frac{2 (5 + 3)}{2} = 8 \\ x y = (\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}) (\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}) = 1 \\ ∴ x^{2} + y^{2} + x y = (x + y)^{2} - x y \\ = 8^{2} - 1 \\ = 63 \end{array}$